Identifier
-
Mp00173:
Integer compositions
—rotate front to back⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤ
Values
[1] => [1] => [[1],[]] => [1] => 1
[1,1] => [1,1] => [[1,1],[]] => [1,1] => 1
[2] => [2] => [[2],[]] => [2] => 1
[1,1,1] => [1,1,1] => [[1,1,1],[]] => [1,1,1] => 1
[1,2] => [2,1] => [[2,2],[1]] => [2,2] => 1
[2,1] => [1,2] => [[2,1],[]] => [2,1] => 1
[3] => [3] => [[3],[]] => [3] => 1
[1,1,1,1] => [1,1,1,1] => [[1,1,1,1],[]] => [1,1,1,1] => 1
[1,1,2] => [1,2,1] => [[2,2,1],[1]] => [2,2,1] => 1
[1,2,1] => [2,1,1] => [[2,2,2],[1,1]] => [2,2,2] => 1
[1,3] => [3,1] => [[3,3],[2]] => [3,3] => 1
[2,1,1] => [1,1,2] => [[2,1,1],[]] => [2,1,1] => 1
[2,2] => [2,2] => [[3,2],[1]] => [3,2] => 1
[3,1] => [1,3] => [[3,1],[]] => [3,1] => 1
[4] => [4] => [[4],[]] => [4] => 1
[1,1,1,1,1] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => [1,1,1,1,1] => 1
[1,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]] => [2,2,1,1] => 1
[1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]] => [2,2,2,1] => 1
[1,1,3] => [1,3,1] => [[3,3,1],[2]] => [3,3,1] => 1
[1,2,1,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => [2,2,2,2] => 1
[1,2,2] => [2,2,1] => [[3,3,2],[2,1]] => [3,3,2] => 1
[1,3,1] => [3,1,1] => [[3,3,3],[2,2]] => [3,3,3] => 1
[1,4] => [4,1] => [[4,4],[3]] => [4,4] => 1
[2,1,1,1] => [1,1,1,2] => [[2,1,1,1],[]] => [2,1,1,1] => 1
[2,1,2] => [1,2,2] => [[3,2,1],[1]] => [3,2,1] => 2
[2,2,1] => [2,1,2] => [[3,2,2],[1,1]] => [3,2,2] => 1
[2,3] => [3,2] => [[4,3],[2]] => [4,3] => 1
[3,1,1] => [1,1,3] => [[3,1,1],[]] => [3,1,1] => 1
[3,2] => [2,3] => [[4,2],[1]] => [4,2] => 1
[4,1] => [1,4] => [[4,1],[]] => [4,1] => 1
[5] => [5] => [[5],[]] => [5] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => 1
[1,1,1,1,2] => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => [2,2,1,1,1] => 1
[1,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [2,2,2,1,1] => 1
[1,1,1,3] => [1,1,3,1] => [[3,3,1,1],[2]] => [3,3,1,1] => 3
[1,1,2,1,1] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [2,2,2,2,1] => 1
[1,1,2,2] => [1,2,2,1] => [[3,3,2,1],[2,1]] => [3,3,2,1] => 2
[1,1,3,1] => [1,3,1,1] => [[3,3,3,1],[2,2]] => [3,3,3,1] => 2
[1,1,4] => [1,4,1] => [[4,4,1],[3]] => [4,4,1] => 1
[1,2,1,1,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [2,2,2,2,2] => 1
[1,2,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => [3,3,2,2] => 1
[1,2,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => [3,3,3,2] => 1
[1,2,3] => [2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => 1
[1,3,2] => [3,2,1] => [[4,4,3],[3,2]] => [4,4,3] => 1
[1,5] => [5,1] => [[5,5],[4]] => [5,5] => 1
[2,1,1,1,1] => [1,1,1,1,2] => [[2,1,1,1,1],[]] => [2,1,1,1,1] => 1
[2,1,1,2] => [1,1,2,2] => [[3,2,1,1],[1]] => [3,2,1,1] => 2
[2,1,2,1] => [1,2,1,2] => [[3,2,2,1],[1,1]] => [3,2,2,1] => 2
[2,1,3] => [1,3,2] => [[4,3,1],[2]] => [4,3,1] => 2
[2,2,1,1] => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => [3,2,2,2] => 1
[2,2,2] => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 2
[2,3,1] => [3,1,2] => [[4,3,3],[2,2]] => [4,3,3] => 2
[2,4] => [4,2] => [[5,4],[3]] => [5,4] => 1
[3,1,1,1] => [1,1,1,3] => [[3,1,1,1],[]] => [3,1,1,1] => 1
[3,1,2] => [1,2,3] => [[4,2,1],[1]] => [4,2,1] => 2
[3,2,1] => [2,1,3] => [[4,2,2],[1,1]] => [4,2,2] => 3
[3,3] => [3,3] => [[5,3],[2]] => [5,3] => 1
[4,1,1] => [1,1,4] => [[4,1,1],[]] => [4,1,1] => 1
[4,2] => [2,4] => [[5,2],[1]] => [5,2] => 1
[5,1] => [1,5] => [[5,1],[]] => [5,1] => 1
[6] => [6] => [[6],[]] => [6] => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1] => 1
[1,1,1,1,1,2] => [1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [2,2,1,1,1,1] => 1
[1,1,1,1,2,1] => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [2,2,2,1,1,1] => 1
[1,1,1,1,3] => [1,1,1,3,1] => [[3,3,1,1,1],[2]] => [3,3,1,1,1] => 3
[1,1,1,2,1,1] => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [2,2,2,2,1,1] => 1
[1,1,1,2,2] => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [3,3,2,1,1] => 3
[1,1,1,3,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [3,3,3,1,1] => 3
[1,1,1,4] => [1,1,4,1] => [[4,4,1,1],[3]] => [4,4,1,1] => 1
[1,1,2,1,2] => [1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [3,3,2,2,1] => 2
[1,1,2,2,1] => [1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [3,3,3,2,1] => 2
[1,1,2,3] => [1,2,3,1] => [[4,4,2,1],[3,1]] => [4,4,2,1] => 2
[1,1,3,2] => [1,3,2,1] => [[4,4,3,1],[3,2]] => [4,4,3,1] => 2
[1,2,1,3] => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => [4,4,2,2] => 3
[1,2,2,2] => [2,2,2,1] => [[4,4,3,2],[3,2,1]] => [4,4,3,2] => 2
[2,1,1,1,1,1] => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => [2,1,1,1,1,1] => 1
[2,1,1,1,2] => [1,1,1,2,2] => [[3,2,1,1,1],[1]] => [3,2,1,1,1] => 2
[2,1,1,2,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [3,2,2,1,1] => 2
[2,1,1,3] => [1,1,3,2] => [[4,3,1,1],[2]] => [4,3,1,1] => 2
[2,1,2,1,1] => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [3,2,2,2,1] => 2
[2,1,2,2] => [1,2,2,2] => [[4,3,2,1],[2,1]] => [4,3,2,1] => 4
[2,1,3,1] => [1,3,1,2] => [[4,3,3,1],[2,2]] => [4,3,3,1] => 3
[2,1,4] => [1,4,2] => [[5,4,1],[3]] => [5,4,1] => 2
[2,2,1,2] => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => [4,3,2,2] => 2
[2,2,2,1] => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => [4,3,3,2] => 2
[2,2,3] => [2,3,2] => [[5,4,2],[3,1]] => [5,4,2] => 2
[2,3,2] => [3,2,2] => [[5,4,3],[3,2]] => [5,4,3] => 2
[3,1,1,1,1] => [1,1,1,1,3] => [[3,1,1,1,1],[]] => [3,1,1,1,1] => 1
[3,1,1,2] => [1,1,2,3] => [[4,2,1,1],[1]] => [4,2,1,1] => 2
[3,1,2,1] => [1,2,1,3] => [[4,2,2,1],[1,1]] => [4,2,2,1] => 2
[3,1,3] => [1,3,3] => [[5,3,1],[2]] => [5,3,1] => 2
[3,2,1,1] => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => [4,2,2,2] => 1
[3,2,2] => [2,2,3] => [[5,3,2],[2,1]] => [5,3,2] => 3
[3,3,1] => [3,1,3] => [[5,3,3],[2,2]] => [5,3,3] => 3
[3,4] => [4,3] => [[6,4],[3]] => [6,4] => 1
[4,1,1,1] => [1,1,1,4] => [[4,1,1,1],[]] => [4,1,1,1] => 1
[4,1,2] => [1,2,4] => [[5,2,1],[1]] => [5,2,1] => 2
[4,2,1] => [2,1,4] => [[5,2,2],[1,1]] => [5,2,2] => 3
[4,3] => [3,4] => [[6,3],[2]] => [6,3] => 1
[5,1,1] => [1,1,5] => [[5,1,1],[]] => [5,1,1] => 1
[5,2] => [2,5] => [[6,2],[1]] => [6,2] => 1
>>> Load all 154 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
outer shape
Description
The outer shape of the skew partition.
Map
rotate front to back
Description
The front to back rotation of the entries of an integer composition.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!